Finite Group Databasebeta

← #27275451

Subgroups — #27275451

Order 13568.

1174 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133196C211yesmaximal, characteristic
2#20274625C211yesmaximal, characteristic
3C2×#16810348C214yesmaximal, complemented
4C53×#1779C211yesmaximal, characteristic
5C23×Dic106C411yescharacteristic
6C2×#13133196—28no—
7C2×#13133196—22no—
8C2×#13133196—22no—
9C22×C4⋊Dic53C411yescharacteristic
10C22×#11719919C2212yes—
11C22×C22⋊Dic53C412yes—
12C2×C22⋊#88322—22no—
13C2×C22⋊#88322C2212yes—
14#16810348—28no—
15C24×C4×C53C2211yescharacteristic
16C2×C22⋊C8×C53—22no—
17C2×C22⋊C8×C53C2212yes—
18C22×Dic106—22no—
19C22×Dic106—24no—
20C22×Dic106—44no—
21C23×Dic53C2×C412yes—
22C23×Dic53C2×C412yes—
23C22×#88322—22no—
24#13133196—216no—
25#13133196—216no—
26C2×C4⋊Dic53—22no—
27C2×C4⋊Dic53—24no—
28C2×#11719919—22no—
29C2×#11719919—22no—
30C2×#11719919—24no—
31C2×#11719919D418yes—
32C2×C22⋊Dic53—24no—
33C2×C22⋊Dic53—24no—
34C2×C22⋊Dic53—24no—
35C22⋊#88322—24no—
36C22⋊#88322—44no—
37C25×C53C2×C411yescharacteristic
38C23×C4×C53—24no—
39C23×C4×C53—44no—
40C23×C4×C53C2×C411yescharacteristic
41C23×C4×C53D412yes—
42C23×C4×C53D412yes—
43C23×C4×C53C2311yescharacteristic
44C22×C8×C53—22no—
45C22⋊C8×C53—24no—
46C22⋊C8×C53—44no—
47C24×C53—24no—
48C24×C53—22no—
49C24×C53—22no—
50C24×C53—22no—
51C24×C53—42no—
52C24×C53C22⋊C412yes—
53C24×C53C22×C411yescharacteristic
54C22×C4×C53—22no—
55C22×C4×C53—24no—
56C22×C4×C53—24no—
57C22×C4×C53—24no—
58C22×C4×C53—24no—
59C22×C4×C53—24no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53—416no—
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D414yes—
68C22×Dic53—22no—
69C22×Dic53—22no—
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22×Dic53—24no—
73C22×Dic53—24no—
74C22×Dic53—44no—
75C22×Dic53C22⋊C414yes—
76C22×Dic53C22⋊C414yes—
77C2×Dic106—28no—
78C2×Dic106—416no—
79C2×Dic106—44no—
80C2×Dic106—44no—
81C2×C8×C53—44no—
82C2×C8×C53—42no—
83C22⋊Dic53—216no—
84C4⋊Dic53—28no—
85#11719919—28no—
86#11719919—28no—
87C2×#88322—44no—
88C2×#88322—42no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—28no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—22no—
95C23×C53—22no—
96C23×C53—22no—
97C23×C53—22no—
98C23×C53—48no—
99C23×C53—48no—
100C23×C53—44no—
101C23×C53—44no—
102C23×C53#10114yes—
103C23×C53C2×C22⋊C412yes—
104C23×C53C2×C22⋊C411yescharacteristic
105C23×C53#13112yes—
106C23×C53#13112yes—
107C2×C4×C53—28no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—44no—
113C2×C4×C53—48no—
114C2×C4×C53—48no—
115C2×C4×C53#9914yes—
116C2×C4×C53#12012yes—
117C2×C4×C53#12012yes—
118C2×Dic53—24no—
119C2×Dic53—24no—
120C2×Dic53—28no—
121C2×Dic53—44no—
122C2×Dic53—44no—
123C2×Dic53—42no—
124C2×Dic53—42no—
125C2×Dic53—44no—
126C2×Dic53—44no—
127Dic106—416no—
128C8×C53—82no—
129#88322—82no—
130#8308—531nomaximal, Sylow
131C22×C53—24no—
132C22×C53—24no—
133C22×C53—22no—
134C22×C53—22no—
135C22×C53—24no—
136C22×C53—24no—
137C22×C53—24no—
138C22×C53—416no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53—42no—
142C22×C53—42no—
143C22×C53#37911yescharacteristic
144C22×C53#40814yes—
145C22×C53#40811yescharacteristic
146C22×C53#40811yescharacteristic
147C22×C53C2×#9912yes—
148C22×C53C2×#10112yes—
149C4×C53—22no—
150C4×C53—22no—
151C4×C53—24no—
152Dic53—44no—
153Dic53—44no—
154C22×C22⋊Q8—531no—
155#1779—531no—
156#1779D5311yescharacteristic
157C2×#324—534no—
158C2×C53—24no—
159C2×C53—44no—
160C2×C53#147212yes—
161C2×C53#147212yes—
162C2×C53#178011yescharacteristic
163C2×C53#178111yescharacteristic
164C2×C53#178411yescharacteristic
165C23×Q8—531no—
166C24×C4C2×D5311yescharacteristic
167C2×C22⋊Q8—1068no—
168C2×C22⋊Q8—1062no—
169C2×C22⋊Q8—1062no—
170C22×C4⋊C4—532no—
171C22×C4⋊C4—531no—
172C22×C22⋊C4—532no—
173C2×C22⋊C8—22no—
174C2×C22⋊C8—532no—
175C2×C22⋊C8—1062no—
176C2×C22⋊C8C2×D5312yes—
177#324—1068no—
178C53#830811yescharacteristic, complemented, Sylow
179C25C4×D5311yescharacteristic
180C22×Q8—1064no—
181C22×Q8—1062no—
182C22×Q8—2124no—
183C23×C4—24no—
184C23×C4—44no—
185C23×C4—532no—
186C23×C4—532no—
187C23×C4C4×D5311yescharacteristic
188C23×C4D21212yes—
189C23×C4C22⋊D5312yes—
190C23×C4C22×D5311yescharacteristic
191C22×C8—22no—
192C22×C8—1062no—
193C22⋊Q8—10616no—
194C22⋊Q8—10616no—
195C2×C4⋊C4—538no—
196C2×C4⋊C4—1064no—
197C2×C4⋊C4—1062no—
198C2×C4⋊C4—1064no—
199C2×C4⋊C4—1062no—
200C2×C4⋊C4—1062no—
201C2×C22⋊C4—1064no—
202C2×C22⋊C4—1064no—
203C2×C22⋊C4—1064no—
204C22⋊C8—24no—
205C22⋊C8—44no—
206C22⋊C8—1064no—
207C22⋊C8—2124no—
208C24—24no—
209C24—22no—
210C24—22no—
211C24—22no—
212C24—42no—
213C24#1171992112yes—
214C24C2×C4×D5311yescharacteristic
215C2×Q8—1068no—
216C2×Q8—21216no—
217C2×Q8—2124no—
218C2×Q8—2124no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—416no—
229C22×C4—534no—
230C22×C4—534no—
231C22×C4—1062no—
232C22×C4—1062no—
233C22×C4—1062no—
234C22×C4—1062no—
235C22×C4—1064no—
236C22×C4—1064no—
237C22×C4—2124no—
238C22×C4#1171992112yes—
239C22×C4C2×D21211yescharacteristic
240C22×C4D4×D5314yes—
241C22×C4C2×C22⋊D5311yescharacteristic
242C2×C8—44no—
243C2×C8—42no—
244C2×C8—2124no—
245C2×C8—2122no—
246C4⋊C4—1068no—
247C4⋊C4—1068no—
248C4⋊C4—1068no—
249C22⋊C4—10616no—
250C23—22no—
251C23—22no—
252C23—28no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—48no—
260C23—48no—
261C23—44no—
262C23—44no—
263C23#1313313514yes—
264C23C22⋊C4×D5312yes—
265C23C2×#1171992111yescharacteristic
266C23#1313322312yes—
267C23#1313322512yes—
268Q8—21216no—
269C2×C4—28no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—44no—
275C2×C4—48no—
276C2×C4—48no—
277C2×C4—1068no—
278C2×C4—1064no—
279C2×C4—1064no—
280C2×C4—2124no—
281C2×C4—2124no—
282C2×C4—2122no—
283C2×C4—2122no—
284C2×C4—2124no—
285C2×C4—2124no—
286C2×C4#1313313114yes—
287C2×C4#1313318512yes—
288C2×C4#1313318712yes—
289C8—82no—
290C8—4242no—
291C22—24no—
292C22—24no—
293C22—22no—
294C22—22no—
295C22—24no—
296C22—24no—
297C22—24no—
298C22—416no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22#1681047811yescharacteristic
304C22#1681058814yes—
305C22#1681058911yescharacteristic
306C22#1681059111yescharacteristic
307C22C2×#1313313112yes—
308C22C2×#1313313512yes—
309C4—22no—
310C4—22no—
311C4—24no—
312C4—2124no—
313C4—2124no—
314C2—24no—
315C2—44no—
316C2#2027307812yes—
317C2#2027308012yes—
318C2#2027462611yescharacteristic
319C2#2027462911yescharacteristic
320C2#2027463811yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1779C2
socleC23×C53C2×C22⋊C4
prime radicalC25×C53C2×C4
cosocleC23×C4×C53C23
hypercenter#1779D53
hypocenterC53#8308
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#16810348C2semidirect
178. C53#8308semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#131331962
#202746252
C2×#168103482
C53×#17792
#830853